Uso de TIC en los negocios

A paper to be submitted to Plant Physiology

K.A. Meade, M. Cooper, W.D. Beavis

Abstract

Canonical models of growth and development were compared to determine which provided the best description of maize kernel biomass accumulation. Observations of kernel dry weights starting shortly after pollination through maturity were regressed onto a measure of thermal time. Observations from differing maize hybrids taken in two years with significantly different weather patterns were used to construct the model. Three criteria were used to select from possible nonlinear growth and development functions. The function has to produce a sigmoid curve, the parameters should have biological meaning in the context of kernel biomass accumulation, and the model should easily converge over a range of growth patterns. Of the five nonlinear functions described, the Weibull and Gompertz functions were found to describe the pattern of biomass accumulation best. The application of a variance assumption was used to account for heteroscedastic errors and comparisons of methods of variance transformation or assumptions are included.

Introduction

Maize kernel development and the accompanying biomass accumulation are complex and multifaceted processes. However, a simple and accurate model of the accumulation of biomass in maize kernels could provide a much needed description of important aspects of kernel

development. The selected model would need to be flexible enough for use in a variety of environments and genotypes, but simple enough that it could be included in a more complex whole-plant model. This paper describes the selection of such a model using a sample of commercial hybrids as an example.

Maize kernels are part of the ear meristem and their growth is initiated as early as the three-leaf stage (Ritchie et al., 2008). The tissues of the ear grow and differentiate, eventually forming many individual female flowers. Each flower, or spikelet, has the potential to become a kernel of grain with a diploid embryo and triploid endosperm. The pistil of the kernel elongates to form the silk which eventually grows out of the husks, modified leaves which surround the ear, and is receptive to pollen and fertilization (Kisselbach, 1949).

The stage of growth when pollen shed begins and silks emerge represents a significant transition in the maize life cycle. No further leaves are formed, and the ear or ears become the major sink, i.e. the primary user of photosynthate (Ritchie et al., 2008). This is also the stage of growth in which kernels can be distinguished and described. Beginning with fertilization, kernels can be measured and evaluated throughout its growth and development. The cells of the embryo and endosperm begin dividing immediately after a double-fertilization event, but little or no biomass is accumulated until ten to fifteen days after pollination. Once biomass accumulation begins, it continues at a nearly constant rate until the kernel starts to approach physiological maturity. The preparation for physiological maturity is evidenced by a slowing of biomass

accumulation, the formation of a layer of dead cells at the abscission zone, and loss of moisture. After physiological maturity, the kernel enters a quiescent state until germination (Kisselbach, 1949).

Kernel growth has been viewed from the perspective of the biochemical reactions within the embryo (Murray, 1988) and the endosperm (Lopes and Larkins, 1993), and from the

perspective of genetic regulation at each stage (Wang et al., 2009). One way to encompass all of these perspectives is through modeling kernel growth. Models of plant growth have been

developed for many species and processes (Thornley and France, 2007). This is because a model of a growth process can provide descriptions and predictions that would be difficult to obtain experimentally. For example, growth measurements are time and resource consuming;

development of a model to describe growth of a plant can save on both. Once a model is in place it is relatively easy to view the effect that changes in environment or genotype may produce, and fewer observations are needed to predict the outcome of such changes (Thornley and France, 2007). Kernel growth has been modeled many times using many different methods (see below). However these models have largely failed to describe biomass accumulation in the kernel using a continuous function. Nonlinear models are often continuous.

One example of a nonlinear function is a sigmoid function which produces an s-shaped curve. Two of the most recognizable sigmoid curves, the logistic and the Gompertz, were initially derived to describe population growth under limiting conditions and human mortality respectively (Verhulst, 1838; Gompertz, 1825). These and other sigmoid functions have since been applied to a wide variety of agricultural processes, particularly physiological processes of plants (Thornley and Johnson, 1990; Peek et al., 2002; Yin et al., 2003).

The accumulation of biomass in a kernel of grain follows a sigmoid pattern of growth. The pattern of sigmoid kernel growth is similar among maize, sorghum (Gambín et al., 2008) and

wheat (Pepler et al., 2006). There is an initial phase, termed the lag phase, in which endosperm cells undergo division during which there is little or no biomass accumulation. The second phase of growth is near-linear with continuous accumulation of dry matter within the endosperm. The final phase is the slowing of growth until a final weight is reached. At some point in the final stage, the kernel reaches physiological maturity and ceases to accumulate biomass (Bewley and Black, 1994; Murray, 1988; Sala et al., 2007; Saini and Westgate, 1999).

Data for biomass accumulation can be obtained through either a destructive or nondestructive method. Ideally a nondestructive method would be used to measure kernel biomass accumulation. This would allow for multiple measurements to be taken on the same individual When a destructive sampling technique is used, multiple individuals must be measured; however variability among individuals’ genetic backgrounds and environmental conditions will increase the overall experimental error (Thornley and France, 2007). A method of estimating moisture percentage of the ear as a whole has been developed with some success (Reid et al., 2010). However, there are multiple obstacles to estimating biomass accumulation of a kernel while it is still attached to the ear. The ear consists of a central cob with rows of kernels attached, and layers of husks surrounding the kernels. Adjacent kernels make estimating an individual kernel weight extremely difficult, and the moisture content must be removed or

estimated before the biomass content can be determined. The husks, modified leaves surrounding the ear, are an important source of photosynthate and damaging the husks may reduce the overall accumulation of biomass in the ear (Murray, 1988). For these reasons, a destructive method involving the harvesting of an ear from the field and removal of kernels from that ear is

commonly used (Borrás et al., 2010). Taking samples from individuals with identical genotypes help to reduce the experimental variability and can be used to construct a model of biomass accumulation.

There are many ways to model biomass accumulation in a kernel of grain. Perhaps the simplest method would be to isolate the samples taken at a point in development and measure the dry weight of the kernel as a phenotypic trait. However, a biomass measurement taken at the end of grain filling will provide no information on grain fill as a process. Measurements from multiple growth stages in a longitudinal design could be analyzed as a multivariate trait, but underlying assumptions about non-correlated errors would be erroneous (Davidian and Giltinan, 2003). Further, information about the underlying process is often lost with this technique (Weiner and Thomas, 1992; Vega and Sadras, 2003).

Another approach is to consider the stage of linear growth and attempt to determine the growth rate of the kernel by determining the slope of the linear regression of dry weight of the kernel onto GDD. This empirical technique has been used for several species including maize, barley, and wheat (Tollenaar and Bruulsema, 1988; Reddy and Daynard, 1983; Jones et al., 1996; Voltas et al., 1999; Ehdaie et al., 2008; Cross, 1975). Determination of the growth rate is of considerable interest when studying source/sink ratios between the plant and the kernel (Jones et al., 1996; Borrás and Otegui, 2001). However measurements taken during the lag phase and after physiological maturity were not considered in these studies. Further, within each study, times were selected to distinguish the beginning and end of the linear phase of growth. Selection of the time at which the linear growth period begins and ends was subjective and differed among studies in maize (Tollenaar and Bruulsema, 1988; Reddy and Daynard, 1983; Jones et al., 1996; Cross, 1975).

The second form of linear model is the segmented model which combines multiple linear models to describe a nonlinear process. This has been by far the most common approach to describe kernel growth. First described as a method to determine the ear fill period and growth rate, the use of a segmented model has been argued as a means to avoid taking observations at the

beginning and end of kernel growth (Egli, 1998). A bilinear model has been used in maize to describe the period of linear growth and the period after the kernel has reached its final weight. (Borrás et al., 2009; Tanaka and Madonni, 2008; Borrás and Otegui, 2001; Gambín et al., 2007; Mayer et al., 2012; Sala et al., 2007; Gambín et al., 2008). A similar approach has been used for sorghum (Gambín et al., 2008) and wheat (Brocklehurst 1977; Pepler et al., 2006). An additional segment has been added to create a tri-linear model that also describes the lag phase for sorghum (Gambín and Borrás, 2007) and wheat (Calderini et al., 2000; Calderini and Reynolds, 2000). A fourth phase was added to a model describing sorghum kernel growth to describe the break approximately midway through the linear growth phase at which growth decreases, but not to the extent of the final phase (Yang et al., 2010). Segmented models are computationally tractable and much less complex than a mechanistic nonlinear model. However this comes at the expense of being highly susceptible to outliers, particularly at the breakpoints where models meet. These models are also highly reliant upon the analyst to provide an appropriate number of segments and positions of breakpoints (Hunt, 1982).

The use of segmented models assumes that each stage of growth is distinct from the next. The available physiology and observations from previous studies suggests that this is not the case. Growth of the embryo and endosperm is continuous throughout kernel development (Nardman and Werr, 2009; Scanlon and Takacs, 2009). There is information about the nature of the biological processes occurring that is lost when growth and development are not considered in a continuous manner (Thornley and France, 2007). For example, segmented models often use the intersection of the second and third segments to indicate physiological maturity (Borrás et al., 2009; Tanaka and Madonni, 2008; Borrás and Otegui, 2001; Gambín et al., 2007; Mayer et al., 2012; Sala et al., 2007; Gambín et al., 2008). This would indicate that growth continues unabated until it abruptly stops and physiological maturity is reached. Studies of programmed cell death and maturation of maize kernels have determined that physiological maturity is not abruptly

reached. Instead embryo cells enter a quiescent state and endosperm cells undergo programmed cell death over a period of time (Egli, 2004, Young and Gallie, 2000; Scanlon and Takacs, 2009). For all of these reasons, a continuous model of kernel growth is needed.

A class of nonlinear functions that is easily related to the linear models is those based on polynomials (Kutner et al., 2005). Polynomials are flexible but the parameter values lack any meaning and are difficult to interpret from a biological perspective (Peek et al., 2002; Lei and Zhang, 2004). A lack of biological meaning negates much of the value of a model. While the model may be used for prediction purposes, there is little information gained about the biological processes that are occurring. The lack of information about the biology means that no knowledge about the physiology is gained. Polynomial models can be prone to over-fitting, and the ability of the model to closely describe the growth curve must be carefully balanced against the need for a simple but more powerful model (Yin et al., 2003). A step-wise regression procedure has been used to select polynomial functions for wheat kernel growth. The model that best fit biomass accumulation in the kernel differed between years suggesting a significant environmental effect on kernel growth (Pržulj and Momčilovič, 2011).

A continuous nonlinear model can be used to describe kernel growth. There are many nonlinear functions that model a sigmoid curve, and several of these can be used to model plant growth with biological meaning (Thornley and France, 2007; Hunt, 1982; Thornley and Johnson, 1990). Nonlinear functions have been used to model kernel growth in multiple species. The logistic function (Melchiori and Caviglia, 2008) and the Weibull function (Sala et al., 2007) have been used to model maize kernel weight as a function of thermal units. The logistic function has also been used to describe the growth pattern of triticales (Santiveri et al., 2002) and wheat (Darroch and Baker, 1990). An important difference between the available sigmoid models is whether or not a symmetric growth curve is forced. The logistic function assumes that the

transition from the lag phase to the near-linear growth phase and from growth to maturity occurs at the same rate (Thornley and Johnson, 1990). Comparing nonlinear models using a data set with observations from all phases of kernel growth after pollination provides a valuable opportunity to study whether or not kernel growth is a symmetric process.

None of the previously described studies attempted to contrast the features among nonlinear models for the description of kernel growth as a function of thermal units. Criteria for such contrasts would include a sigmoid function to allow for smooth transitions between growth phases and the parameters would have biological meaning in terms of kernel growth. The research reported herein was conducted to identify a parsimonious and powerful nonlinear function to model maize kernel growth as a function of thermal units. Five nonlinear functions were evaluated for biomass accumulation as a function of thermal units. Statistical procedures for the comparison of nonlinear models are described as well as the incorporation of a variance function to account for heterogeneity of variances. A final model is presented that incorporates a nonlinear regression of kernel weight into a model with a fixed effect of genotype and a variance function to account for heterogeneity of variances.

Materials and methods

Field and entry description

The experiment was conducted over two years, 2009 and 2010 in Dallas Center, Iowa. For purposes of formulating a robust model to describe the variability in kernel biomass accumulation, developing ears were harvested throughout the grain filling period in plots

consisting of five hybrids in 2009 and ten hybrids in 2010. Each plot was 5.4m with a row width of 76.2cm. Planting rate was 32 seeds per plot. The plots were replicated five times in 2009, where the first four replications were sampled, with the fifth replication being used when one of

the first four replications had fewer than 21 plants. In 2010 each two-row plot was replicated three times, and plants from all three replications were harvested. The experimental design was a randomized complete block design in both years. The planting rate was 64 seeds per plot. Row length and spacing in 2010 was the same as in 2009. All hybrids tested were 111 comparative relative maturity (CRM) hybrids provided by DuPont Pioneer (Table 1).

Calculation of growing degree days

Growing degree days (GDD) are measured in thermal units. This method of measurement was developed to improve accuracy in estimation of physiological maturity in maize hybrids 40 years ago and continues to be the standard (Cross and Zuber, 1972; Sprague and Dudley, 1988). Weather information throughout the growing season was collected at Dallas Center, Iowa. Maximum (Max) and minimum (Min) temperatures in degrees Fahrenheit were used in the calculation of growing degree days for each day as:

𝐺𝐷𝐷 = (𝑀𝑎𝑥 + 𝑀𝑖𝑛)₂ − 50

(Cross and Zuber, 1972). A late planting date and early frost resulted in a shortened growing season for 2009. In addition, temperatures were below average for Iowa in 2009. The environment was significantly different for the 2010 growing season. Rainfall was above average, while temperatures were near average and resulted in a full growing season for 111 CRM hybrids. Historic average temperatures and rainfall are available at Mesonet

(http://mesonet.agron.iastate.edu).

Kernel dry weight evaluation over the growing season

Biomass accumulation was determined from kernel dry weights collected throughout the growing season starting shortly after the appearance of silk. Shortly before flowering, plants

within a plot were individually labeled. Silk notes were taken on a per plant basis, and sampling was started approximately four days after 50% of the plants in a plot had silked. The date on which the top ear of a plant had visible silks was noted. The accumulated GDD from silk to harvest of an ear were calculated starting on the day the silks emerged from the ear until the day that the ear was harvested.

An ear was harvested from a plot once every three days. The plant label was taken with the ear to identify the ear and when it was harvested. The GDD that had accumulated between silk and harvest were calculated. Ears from plants that were root lodged or did not have neighboring plants within the row were noted and discarded because kernel biomass

accumulation would have been biased (Duvick and Cassman, 1999). Approximately eighty and sixty samples were taken for each entry in 2009 and 2010 respectively.

After harvesting the ear, fifteen kernels were excised and placed in a pre-weighed vial Kernels were sampled from rows ten to fifteen above the base of the ear; the glumes and pedicel were removed. The samples were dried in a 70ᵒC oven for 96 hours, after which dried kernels were weighed to determine the dry weight of the sample. The kernel dry weight was determined by subtracting the initial vial weight and dividing by 15 to determine the milligrams of dry weight per kernel at harvest (Borrás et al., 2002).

Evaluation of biomass accumulation models

Biomass accumulation in maize kernels results in an s-shaped curve that can be modeled using a nonlinear growth function with growing degree days (GDD) as the predictor, and dry weight as the response. The initial step of selecting a nonlinear function to model maize kernel growth is the selection of several possible equations that fit the basic shape with parameters that can be biologically interpreted. Possible functions were screened based on three criteria: 1) the

function produces a sigmoid curve, 2) the parameters have biological meaning for biomass accumulation in maize kernels and 3) the model should be computationally tractable, i.e., computations should converge on estimates of the parameters. Based on these criteria five possible models were evaluated: Logistic, Gompertz, Chanter, Weibell and Richards. There are multiple parameterizations of these models.

The most basic is the logistic equation (Verhulst, 1838; Yin et al., 2003). The logistic produces a symmetric, s-shaped curve.

𝑊 = 𝑊𝑓−𝑊0

1+𝑒−(𝑔−𝑀)/𝑠 (Eq. 1)

There are many different possible parameterizations of the logistic curve. However, the basic framework included parameters that describe the upper and lower asymptote (W0 and Wf respectively), a scalar parameter with limited biological meaning (s), and a parameter that estimates the GDD (g) at which the inflection point of the curve is reached (M). In addition, the upper asymptote can be halved to determine the weight at the inflection point. A three-parameter logistic was also considered in which the lower asymptote was set to zero

𝑊 = _{1 + 𝑒−(𝑔−𝑀)/𝑠}𝑊𝑓 (Eq. 2) The logistic function meets the requirements of irreversible growth, substrate is proportional to the rate of the biochemical reactions, and the dry weight (amount of biomass) is proportional to the enzyme quantity involved in the biochemical reactions. It is one level of complexity over the simple exponential growth and monomolecular functions where the biochemical reactions occur at a constant rate and substrate is unlimited respectively (Thornley and Johnson, 1990).

𝑊 = 𝑊𝑓𝑒−𝑒−𝑘(𝑔−𝑀) (Eq. 3)

The Gompertz is similar to the logistic function in that it produces an s-shaped curve. However, substrate is assumed to be unlimited and biochemical reaction rates are reduced as GDD

accumulate (Thornley and Johnson, 1990). This results in the possibility of an asymmetric curve and increased flexibility over the logistic, but with a fixed inflection point. There are three